In wall-bounded buoyancy-driven turbulence, the boundary conditions play a crucial role in the flow structure and the global transport property of the system. Bubbles attached to the wall affect these boundary conditions. They often occur in various industrial applications. One example is water electrolysis, where bubbles are generated at the electrodes and can significantly reduce the global mass transport of the system by reducing the active electrode area [1]}, [2]}, [3]}, [4]}, leading to the decrease of the electrolyser efficiency. Another example is catalysis, where bubbles are generated by chemical reactions and can block the catalytic surface, thus also reducing the mass transport [5]}, [6]}, [7]}. One example from daily life is heating water. When a pot of water is heated from below, many tiny gas bubbles nucleate at the bottom wall, reducing the heat transfer efficiency of the system due to the much lower thermal conductivity of gas as compared to the liquid. In all of these examples, bubbles attached to the wall influence the boundary layer (BL) and thus affect the global heat or mass transfer of the system. Therefore, it is highly desirable to quantitatively understand how much the global transport properties are changed. This is the motivation for our study.
